*X*, a sequence is an ordered list of elements of

*X*{x1, x2, x3, ...}. If the number of elements of the sequence is finite, the sequence is referred to as a "finite" sequence. A sequence can also be infinite.

More formally, a sequence is, strictly speaking, a function from some subset of the natural numbers

*N*into some set*X*:

But wait a moment! What if we want for our function to range over a number of values which is greater than that in

*N*? What if we want an*uncountable*sequence of numbers? What would that even mean?!
It turns out that mathematics

*does*in fact give the machinery needed to handle this, although one must dig into set theory to find it. The notion of**Transfinite Sequences**allows for a sequence to be constructed whose length is that of*any*ordinal on*any*well-ordered set. If we accept the Axiom of Choice, then the Well Ordering Theorem holds (or visa versa; the axiom and the theorem are in fact logically equivalent) and we can produce a so-called "transfinite sequence" on*any*set, including the reals. The notions of**Transfinite Induction**and**Transfinite Recursion**also help to properly extend their respective notions which are classically limited to sets of countable order. Pretty cool stuff if you think about exactly what a sequence really is. Also sheds some light on just how extreme the consequences of the Axiom of Choice really are. On a side-note, I have only just begun reading on the topic of transfinite sequences today, so if something in this post is inaccurate, please feel free to correct me.